## You are here

Homecharacterization of field

## Primary tabs

# characterization of field

###### Proposition 1.

Let $\mathcal{R}\neq 0$ be a commutative ring with identity. The ring $\mathcal{R}$ (as above) is a field if and only if $\mathcal{R}$ has exactly two ideals: $(0),\mathcal{R}$.

###### Proof.

($\Rightarrow$) Suppose $\mathcal{R}$ is a field and let $\mathcal{A}$ be a non-zero ideal of $\mathcal{R}$. Then there exists $r\in\mathcal{A}\subseteq\mathcal{R}$ with $r\neq 0$. Since $\mathcal{R}$ is a field and $r$ is a non-zero element, there exists $s\in\mathcal{R}$ such that

$s\cdot r=1\in\mathcal{R}$ |

Moreover, $\mathcal{A}$ is an ideal, $r\in\mathcal{A},s\in\mathcal{S}$, so $s\cdot r=1\in\mathcal{A}$. Hence $\mathcal{A}=\mathcal{R}$. We have proved that the only ideals of $\mathcal{R}$ are $(0)$ and $\mathcal{R}$ as desired.

($\Leftarrow$) Suppose the ring $\mathcal{R}$ has only two ideals, namely $(0),\mathcal{R}$. Let $a\in\mathcal{R}$ be a non-zero element; we would like to prove the existence of a multiplicative inverse for $a$ in $\mathcal{R}$. Define the following set:

$\mathcal{A}=(a)=\{r\in\mathcal{R}\mid r=s\cdot a,\text{ for some }s\in\mathcal{R}\}$ |

This is clearly an ideal, the ideal generated by the element $a$. Moreover, this ideal is not the zero ideal because $a\in\mathcal{A}$ and $a$ was assumed to be non-zero. Thus, since there are only two ideals, we conclude $\mathcal{A}=\mathcal{R}$. Therefore $1\in\mathcal{A}=\mathcal{R}$ so there exists an element $s\in\mathcal{R}$ such that

$s\cdot a=1\in\mathcal{R}$ |

Hence for all non-zero $a\in\mathcal{R}$, $a$ has a multiplicative inverse in $\mathcal{R}$, so $\mathcal{R}$ is, in fact, a field. ∎

## Mathematics Subject Classification

12E99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

## Attached Articles

## Corrections

Proof included by Larry Hammick ✓

assumption by matte ✓

could be more general by scineram ✘